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On the C1-norm of the Hermite interpolation operator
JOURNAL
OF APPROXIMA~rlON
THEORY
50.
160- 166 ( 1987 )
On the Cl-Norm Interpolation BERND
of the Hermite Operator
STEINHAUS
Department qf Mathemaiics, University of Duisburg, 4100 Duisburg, West Germany Communicated by Oved Shish0 Received July 24, 1984: revised March 19, 1985 DEDICATED
TO THE MEMORY OF C&ZA FREUD
1. INTRODUCTION
Consider the interval Z= [ - 1, I] and the space C’(Z) consisting of the continuously differentiable real-valued functions on I provided with the Cl-norm
(1)
llfll, :=max{Il.fll~ llf’ll1~
where /I. 11denotes the Chebyshev norm on 1 and f’ is the first derivative of f~ C’(Z). We use the Chebyshev nodes of the first kind 2i+ 1 xi := cos JJi := cos 2n n,
O
(2)
to construct a Hermite interpolation operator H,: C’(Z)+L’~,+,.
(3)
Pottinger examined the operator norm of H, induced by (1):
IlH,lI~ :=max{ SUP IIKf’ll~ SUP IIWJ)‘ll } c;g’Pl Krcr =: max{ IIH,ll,*~IIHAI: 1. He proved
itm, 32~-4 c419 IIff,//,=~(~)(n+~) IIH,II,* d 5,
C%61.
In the present paper we give the exact rate of growth of 1)H,lI , ; namely, we prove the 160 0021-9045/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
161
Cl-NORM OF HERMITEINTERPOLATION
THEOREM. For the Cl-norm of the Hermite interpolation operator H, based on the zeros of the Chebyshev polynomials of the first kind, the following inequality holds:
l)+fln(n+
d IIH,lll d4(n+
4(n+ I)-5+-&
l)+ 13.
2. SOMEESTIMATES The Hermite interpolation operator (3) is given by Hnf(x)=
i
~;(X)lf(X)f(Xi)+
f
(~X-Xz)C(X)f’(Xi)
i=o
i=O
with Ui(X) = 1 - =(x-xi),
Odidn,
I
and
liCX)= (x -x;) w’(x,)’
OliGn,
where o(x) := n;=,(x - xi) denotes the nodal polynomial. Let us first mention a result of FejCr [Z]: LEMMA 1. For the functions given by (4), we have
i;. u,(x)= (n + 1)‘. From now on we use only the special nodes (2) for which ui(x)= 1 --A I,(X) =s
I
(x-x,,=*>
‘sin yicos(n+ cos y-cos
1 -xxI 1) y yi ’
0 6 i 6 n, where x = cos y, y E [0, n]. Using Lemma 1, we have
(4)
162
UERND STEINHAUS
COROLLARY 1.
For the Chebyshev nodes of’ the ,first kind, MY huve (6) (7)
Prooj: Formula (6) follows immediately from (5) and Lemma 1 for x = - 1. To prove (7), we take I = q(x) = H,,Q(x) and find
Using (6), we get the desired result. LEMMA 2. C(I) below:
The following
estimates hold for the functions
A,(x) := i sin’~~j I=0 llA,ll <(n+
l)+
) ‘sin 2 -ln(n+ 71
B,,(x) := i 7co’ ” (x -xi) ,=. sin I’, 2 )IB,lI <--ln(n+ n
1
y,l;(x) l)+
A,,, B,, C, E
,
1 ;
(8)
If(x) ,
I)+ 1;
(9)
sin(n + 1) y 1 - cos y cos yi cos(n + 1) y , cH(x) ‘= ,co (n + 1) sin y cos y - cos y, llC,j/ d (n + 1) + 2 i ln(n + 1) + 11. Proof:
As A,(x,)=
1, O
(10)
let xfx,.
Case 1. If sin y < (n + 1) ‘j2, then A,(x) ,< (n + 1)“2 i /l,(x)/ d (n + 1)‘12 aln(n+ I=0
l)+
1
Concerning the estimate of the Lebesgue constant, see Rivlin [7].
Cl-NORM
Case 2.
OF HERMITE
163
INTERPOLATION
For sin y > (n + 1) P”2, we get (x # x,) n Isin(n+l)yl ,c sin y 1-O
An(x) =A 6-
cos(n + 1) y cos y-cos yi
sin2 yi
Isin(n + 1) yl (sin yi [sin y, - sin yI l i sin y n+ 1 j=o + sin yi sin y)
< f
sin y, - sin y
sin yi
cos y-cos y,
i=O
< f
cos(n + 1) y cos y - cos y,
(sin y, + sin y)
i=O
+ i
I/i(X)1
i=O
‘ln y’-‘ln ’ + f Il,(x)l cos y - cos y; i=o
y + cos yil f i =;co Icos
I~,(-~)1
< (n + 1) + 2 ln(n + l)+l i 7t
I
r=O
AS (n + 1) lsin yil 3 (n + 1) sin y, 3 (n + 1)(2/n) y. = 1, we obtain (9): 1 i ll,(x)l
B’(x)6
l)+
1.
To prove (lo), we substitute l-cos
ycos y,=sin2
y+cos2y-cos
ycos yi-sin
ysin yi
+ sin y sin yI = sin y(sin y-sin
yi) + cos y(cos y - cos yi)
+ sin y sin yi. By the triangle inequality C,(x)<
i
I/,(x)1
+ i i=O
< i i=O
+
i i=o
i=O
Ismn+
l)
yI
lcos
y cos(n+ 1) yl
(n+l)siny
Isin(n + 1) y cos(n + 1) yl 1sin y-sin cos y-cos n+l
[l;(x)1 + i 1 +Jn+l ,=o
n cot y r=o xl
.
yj y;
164 Combining
BERND STEINHAUS
this with the inequality
(cf. Ehlich and Zeller [ 1] ) yields the desired result.
0
3. PROOF OF THE THEOREM
We first prove the lower estimate. To this end, we take a sufficiently small E> 0 and a function f;: E C’(I) with the properties ,fi:(x,) = -x,
Odidn;
- ic,
.I”:(-~,) = 13
O
IlLIll = 1. Then
=
i (-x,-k) i=O
+il
[{
2px,
i
1=0
2+
&IT+
x,
(n+
sin2 yi (n+ 1)2(1 -x,)~ II
=~++&}~o&+&.s& +2
k (1+x,),=O
l (12+1)2o
n 1+x, 1 -xi
=~&~+{2+&}~o(l-&) 3
-(n
l-x 1-i 1 ;=o(l -xJ2 i
--L +2(n+1)+(n+*)2i=0
i
l
;=Ji-Fg (l-&)1.
I ,
1)2(1 3 -xi)*
I
165 Let E+ O+. By Corollary 1 ll4{2+~}
Un+1Hn+W
-&
i
(n+l)‘-;(n+l)‘;(n+l)‘}
l)+ j-&y
+2(n+
un+
1)-2(n+
4
2 n+l
=4(n+l)-5+-.
To prove the upper estimate, we use Pottinger’s relation [S, 61 (H,f)‘(x)
= i
{u;(x) If(x) + 211,(x) f,(x) f:(x)} j-‘fr(s) 1
i=O
+ f
ds
{f?(x) + 2(x - x;) f,(x) f:(x)} f’(xJ.
i=O
Let f~ C’(Z) satisfy (/f 11,= 1, so that 11 f'II < 1. By Lemma 2,
I(Hrzf )‘(x)l 6 i
I {4(x) 8x1 + 2u,(x) I;
uxw-
x,)1
i=O
+ ice Iax) + 2(x - x,1 l,(x) Mx)l n =
i=.
Z&$
C(x)-2[?(x)
CxAxi) I
Zi
+2
sin(n + 1) y 1 - cos y cos yi cos(n + 1) y (n+l)siny cosy-cosy, -f?(x)
+i i=O
+2
sin y, sin(n + 1) y C(x) sin y
< 2A”(X) + B,(x) + 2C,(x) + 3 5 f?(x) ,=O
<4(n+1)+7
qln(n+l)+l { IT
I
+6,
since Cyzo If(x) d 2. The proof is complete since IIZZ,ll$ d 5 (Pottinger
C%61). I
166
BERND STEINHAUS REFERENCES
1. H. EHL~H AND K. ZELLER, Auswertung dcr Normen von Interpolationsoperatoren, Mar/t. Ann. 164 (1966) 105-l 12. 2. L. FEJ~R, liber einige Identitaten der Intcrpolationstheorie und ihre Anwendungen zur Bestimmung kleinster Maxima, Arta Sci. Math. 5 (1932) 1455153. 3. I. P. NATANSON. “Constructive Function Theory,” Vol. Ill, Ungar, New York, 1965. 4. P. POTTINGER, Zur Hermite-Interpolation, Z. Angrn’. Math. Mech. 56 (1976), T31GT311. im Raum @(l),” Habilitationsschrift, 5. P. POTTINGER.“Zur linearen Approximation Duisburg, 1976. 6. P. POTTINGER,On the approximation of functions and their derivatives by Hermite interpolation, .f. Approu. Theory 23 (1978) 2677273. 7. T. J. RIVLIN, “The Chebyshev Polynomials,” Wiley, New York, 1974.
OF APPROXIMA~rlON
THEORY
50.
160- 166 ( 1987 )
On the Cl-Norm Interpolation BERND
of the Hermite Operator
STEINHAUS
Department qf Mathemaiics, University of Duisburg, 4100 Duisburg, West Germany Communicated by Oved Shish0 Received July 24, 1984: revised March 19, 1985 DEDICATED
TO THE MEMORY OF C&ZA FREUD
1. INTRODUCTION
Consider the interval Z= [ - 1, I] and the space C’(Z) consisting of the continuously differentiable real-valued functions on I provided with the Cl-norm
(1)
llfll, :=max{Il.fll~ llf’ll1~
where /I. 11denotes the Chebyshev norm on 1 and f’ is the first derivative of f~ C’(Z). We use the Chebyshev nodes of the first kind 2i+ 1 xi := cos JJi := cos 2n n,
O
(2)
to construct a Hermite interpolation operator H,: C’(Z)+L’~,+,.
(3)
Pottinger examined the operator norm of H, induced by (1):
IlH,lI~ :=max{ SUP IIKf’ll~ SUP IIWJ)‘ll } c;g’Pl Krcr =: max{ IIH,ll,*~IIHAI: 1. He proved
itm, 32~-4 c419 IIff,//,=~(~)(n+~) IIH,II,* d 5,
C%61.
In the present paper we give the exact rate of growth of 1)H,lI , ; namely, we prove the 160 0021-9045/87 $3.00 Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
161
Cl-NORM OF HERMITEINTERPOLATION
THEOREM. For the Cl-norm of the Hermite interpolation operator H, based on the zeros of the Chebyshev polynomials of the first kind, the following inequality holds:
l)+fln(n+
d IIH,lll d4(n+
4(n+ I)-5+-&
l)+ 13.
2. SOMEESTIMATES The Hermite interpolation operator (3) is given by Hnf(x)=
i
~;(X)lf(X)f(Xi)+
f
(~X-Xz)C(X)f’(Xi)
i=o
i=O
with Ui(X) = 1 - =(x-xi),
Odidn,
I
and
liCX)= (x -x;) w’(x,)’
OliGn,
where o(x) := n;=,(x - xi) denotes the nodal polynomial. Let us first mention a result of FejCr [Z]: LEMMA 1. For the functions given by (4), we have
i;. u,(x)= (n + 1)‘. From now on we use only the special nodes (2) for which ui(x)= 1 --A I,(X) =s
I
(x-x,,=*>
‘sin yicos(n+ cos y-cos
1 -xxI 1) y yi ’
0 6 i 6 n, where x = cos y, y E [0, n]. Using Lemma 1, we have
(4)
162
UERND STEINHAUS
COROLLARY 1.
For the Chebyshev nodes of’ the ,first kind, MY huve (6) (7)
Prooj: Formula (6) follows immediately from (5) and Lemma 1 for x = - 1. To prove (7), we take I = q(x) = H,,Q(x) and find
Using (6), we get the desired result. LEMMA 2. C(I) below:
The following
estimates hold for the functions
A,(x) := i sin’~~j I=0 llA,ll <(n+
l)+
) ‘sin 2 -ln(n+ 71
B,,(x) := i 7co’ ” (x -xi) ,=. sin I’, 2 )IB,lI <--ln(n+ n
1
y,l;(x) l)+
A,,, B,, C, E
,
1 ;
(8)
If(x) ,
I)+ 1;
(9)
sin(n + 1) y 1 - cos y cos yi cos(n + 1) y , cH(x) ‘= ,co (n + 1) sin y cos y - cos y, llC,j/ d (n + 1) + 2 i ln(n + 1) + 11. Proof:
As A,(x,)=
1, O
(10)
let xfx,.
Case 1. If sin y < (n + 1) ‘j2, then A,(x) ,< (n + 1)“2 i /l,(x)/ d (n + 1)‘12 aln(n+ I=0
l)+
1
Concerning the estimate of the Lebesgue constant, see Rivlin [7].
Cl-NORM
Case 2.
OF HERMITE
163
INTERPOLATION
For sin y > (n + 1) P”2, we get (x # x,) n Isin(n+l)yl ,c sin y 1-O
An(x) =A 6-
cos(n + 1) y cos y-cos yi
sin2 yi
Isin(n + 1) yl (sin yi [sin y, - sin yI l i sin y n+ 1 j=o + sin yi sin y)
< f
sin y, - sin y
sin yi
cos y-cos y,
i=O
< f
cos(n + 1) y cos y - cos y,
(sin y, + sin y)
i=O
+ i
I/i(X)1
i=O
‘ln y’-‘ln ’ + f Il,(x)l cos y - cos y; i=o
y + cos yil f i =;co Icos
I~,(-~)1
< (n + 1) + 2 ln(n + l)+l i 7t
I
r=O
AS (n + 1) lsin yil 3 (n + 1) sin y, 3 (n + 1)(2/n) y. = 1, we obtain (9): 1 i ll,(x)l
B’(x)6
l)+
1.
To prove (lo), we substitute l-cos
ycos y,=sin2
y+cos2y-cos
ycos yi-sin
ysin yi
+ sin y sin yI = sin y(sin y-sin
yi) + cos y(cos y - cos yi)
+ sin y sin yi. By the triangle inequality C,(x)<
i
I/,(x)1
+ i i=O
< i i=O
+
i i=o
i=O
Ismn+
l)
yI
lcos
y cos(n+ 1) yl
(n+l)siny
Isin(n + 1) y cos(n + 1) yl 1sin y-sin cos y-cos n+l
[l;(x)1 + i 1 +Jn+l ,=o
n cot y r=o xl
.
yj y;
164 Combining
BERND STEINHAUS
this with the inequality
(cf. Ehlich and Zeller [ 1] ) yields the desired result.
0
3. PROOF OF THE THEOREM
We first prove the lower estimate. To this end, we take a sufficiently small E> 0 and a function f;: E C’(I) with the properties ,fi:(x,) = -x,
Odidn;
- ic,
.I”:(-~,) = 13
O
IlLIll = 1. Then
=
i (-x,-k) i=O
+il
[{
2px,
i
1=0
2+
&IT+
x,
(n+
sin2 yi (n+ 1)2(1 -x,)~ II
=~++&}~o&+&.s& +2
k (1+x,),=O
l (12+1)2o
n 1+x, 1 -xi
=~&~+{2+&}~o(l-&) 3
-(n
l-x 1-i 1 ;=o(l -xJ2 i
--L +2(n+1)+(n+*)2i=0
i
l
;=Ji-Fg (l-&)1.
I ,
1)2(1 3 -xi)*
I
165 Let E+ O+. By Corollary 1 ll4{2+~}
Un+1Hn+W
-&
i
(n+l)‘-;(n+l)‘;(n+l)‘}
l)+ j-&y
+2(n+
un+
1)-2(n+
4
2 n+l
=4(n+l)-5+-.
To prove the upper estimate, we use Pottinger’s relation [S, 61 (H,f)‘(x)
= i
{u;(x) If(x) + 211,(x) f,(x) f:(x)} j-‘fr(s) 1
i=O
+ f
ds
{f?(x) + 2(x - x;) f,(x) f:(x)} f’(xJ.
i=O
Let f~ C’(Z) satisfy (/f 11,= 1, so that 11 f'II < 1. By Lemma 2,
I(Hrzf )‘(x)l 6 i
I {4(x) 8x1 + 2u,(x) I;
uxw-
x,)1
i=O
+ ice Iax) + 2(x - x,1 l,(x) Mx)l n =
i=.
Z&$
C(x)-2[?(x)
CxAxi) I
Zi
+2
sin(n + 1) y 1 - cos y cos yi cos(n + 1) y (n+l)siny cosy-cosy, -f?(x)
+i i=O
+2
sin y, sin(n + 1) y C(x) sin y
< 2A”(X) + B,(x) + 2C,(x) + 3 5 f?(x) ,=O
<4(n+1)+7
qln(n+l)+l { IT
I
+6,
since Cyzo If(x) d 2. The proof is complete since IIZZ,ll$ d 5 (Pottinger
C%61). I
166
BERND STEINHAUS REFERENCES
1. H. EHL~H AND K. ZELLER, Auswertung dcr Normen von Interpolationsoperatoren, Mar/t. Ann. 164 (1966) 105-l 12. 2. L. FEJ~R, liber einige Identitaten der Intcrpolationstheorie und ihre Anwendungen zur Bestimmung kleinster Maxima, Arta Sci. Math. 5 (1932) 1455153. 3. I. P. NATANSON. “Constructive Function Theory,” Vol. Ill, Ungar, New York, 1965. 4. P. POTTINGER, Zur Hermite-Interpolation, Z. Angrn’. Math. Mech. 56 (1976), T31GT311. im Raum @(l),” Habilitationsschrift, 5. P. POTTINGER.“Zur linearen Approximation Duisburg, 1976. 6. P. POTTINGER,On the approximation of functions and their derivatives by Hermite interpolation, .f. Approu. Theory 23 (1978) 2677273. 7. T. J. RIVLIN, “The Chebyshev Polynomials,” Wiley, New York, 1974.